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XI.—Theory of Indices for Non-Associative Algebra*

Published online by Cambridge University Press:  14 February 2012

I. M. H. Etherington
I. M. H. Etherington
Affiliation:
Mathematical Institute, University of Edinburgh.
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Synopsis

The logarithmetic L of a non-associative algebra or class of algebras S has been previously defined as the arithmetic of the indices of powers of the general element when indices are added (non-associatively) and multiplied by certain conventions similar to those of ordinary algebra. With respect to addition, L is a homomorphic image of the “most general” logarithmetic B, the free additive groupoid with one generator 1, and in the case of algebras of one operation is essentially the same as the free algebra in one variable on S. The definition is now extended so that L is defined when S is any subset of an algebra or class of subsets of algebras, with the result that every homomorph of B is a logarithmetic ; but a distinction has then to be drawn between closed logarithmetics in which as before both addition and multiplication are defined, and other logarithmetics in which there is only addition. L is its own logarithmetic (taken with respect to addition) only if L is closed. For subsets of palintropic algebras, L is necessarily closed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 64 , Issue 2 , 1954 , pp. 150 - 160
Copyright
Copyright © Royal Society of Edinburgh 1954

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References

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